Problem: Determine how many solutions exist for the system of equations. ${4x-2y = -8}$ ${-6x+3y = -9}$
Convert both equations to slope-intercept form: ${4x-2y = -8}$ $4x{-4x} - 2y = -8{-4x}$ $-2y = -8-4x$ $y = 4+2x$ ${y = 2x+4}$ ${-6x+3y = -9}$ $-6x{+6x} + 3y = -9{+6x}$ $3y = -9+6x$ $y = -3+2x$ ${y = 2x-3}$ Just by looking at both equations in slope-intercept form, what can you determine? ${y = 2x+4}$ ${y = 2x-3}$ Both equations have the same slope with different y-intercepts. This means the equations are parallel. ${1}$ ${2}$ ${3}$ ${4}$ ${5}$ ${6}$ ${7}$ ${8}$ ${9}$ ${\llap{-}2}$ ${\llap{-}3}$ ${\llap{-}4}$ ${\llap{-}5}$ ${\llap{-}6}$ ${\llap{-}7}$ ${\llap{-}8}$ ${\llap{-}9}$ ${1}$ ${2}$ ${3}$ ${4}$ ${5}$ ${6}$ ${7}$ ${8}$ ${9}$ ${\llap{-}2}$ ${\llap{-}3}$ ${\llap{-}4}$ ${\llap{-}5}$ ${\llap{-}6}$ ${\llap{-}7}$ ${\llap{-}8}$ ${\llap{-}9}$ Parallel lines never intersect, thus there are NO SOLUTIONS.